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This school consists of an array of courses which at first glance may seem to have little in common. The underlying structure relating gauge theory to enumerative geometry to number theory is string theory. In this short introduction, we will attempt to give a schematic overview of how the various topics covered in this school fit into this overarching framework.

81T30 ; 83E30

The genus-two Kawazumi-Zhang (KZ) invariant is a real-analytic modular function on the Siegel upper half-plane of degree two, which plays an important role in arithmetic geometry. In String theory, it appears as part of the integrand in two-loop four-graviton scattering amplitudes. With hindsight from String theory, I will show that the KZ invariant can be obtained as a generalized Borcherds lift from a weak Jacobi form of index 1 and weight 2. This implies that the KZ invariant is an eigenmode of the quadratic and quartic Casimir operators, and gives access to the full asymptotic expansion in all possible degeneration limits. It also reveals a mock-type holomorphic Siegel modular form underlying the KZ invariant. String theory amplitudes involves modular integrals of the KZ invariant (times lattice partition functions) on the Siegel upper half-plane, which provide new examples of automorphic objects on orthogonal Grassmannians, beyond the usual Langlands-Eisenstein series.

Note: this talk is based on the preprint  "A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces" available on arXiv:1504.04182. Following up on a question asked during the talk (which was answered very poorly), the author obtained shortly after a proof of the conjecture stated in this preprint and during the talk. The proof is available in the revised version arXiv:1504.04182v2.
The genus-two Kawazumi-Zhang (KZ) invariant is a real-analytic modular function on the Siegel upper half-plane of degree two, which plays an important role in arithmetic geometry. In String theory, it appears as part of the integrand in two-loop four-graviton scattering amplitudes. With hindsight from String theory, I will show that the KZ invariant can be obtained as a generalized Borcherds lift from a weak Jacobi form of index 1 and weight 2. ...

81T30 ; 83E30

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